communicoupling
You prefer the beach to the mountains, the mountains to the city, and the city to the beach — and each comparison, taken alone, is honest. A hiring committee prefers candidate A to B, B to C, and C to A, every member perfectly rational. In 1945 Warren McCulloch found the same circle wired into nervous systems and read it as a discovery: a net of as few as six neurons can prefer A over B, B over C and C over A — a pattern no single scalar value could produce — so what an organism values is determined by the topology of its network, not by a magnitude being maximised. He called it a heterarchy of values.
The social twin came first. In 1785 Condorcet showed that majority voting can take perfectly transitive individuals and return an intransitive group: every voter holds a ladder, the committee holds a circle. And a circle has a sting — once preferences cycle, the order of the pairwise votes decides the result, so whoever controls the agenda controls the outcome. Rank still exists locally, in every head-to-head; it fails to exist globally. David Stark later gave the pattern an organisational reading: systems that keep several orders of worth alive trade coherence for adaptability.
Below, two models. A committee of real rankings votes every pair; when its majorities circle, an agenda machine lets you seat any winner you like. Beside it, McCulloch's six-unit circuit answers each pairwise question by wiring alone — no number anywhere in the mechanism.
The mechanism made exact
McCulloch's argument is a short proof. Suppose choice were driven by one value scale u: then u(A) > u(B) > u(C) > u(A), which is impossible. So any creature that reliably picks A over B, B over C and C over A cannot be maximising a single magnitude. He then built the counterexample — a nervous net of six neurons, three drives crossed by three inhibitory links, whose pairwise behaviour is exactly that circle. Engine 2 is his construction in miniature: present any pair and one drive silences the other through its interneuron; chase the mechanism round and the third link closes the loop. What the organism values is written in the wiring diagram, and the wiring need not spell a ladder. He named such systems heterarchies.
Condorcet found the social copy. Each voter in engine 1 holds a strictly transitive ranking; the matrix tallies every head-to-head; and in diverse committees the majorities circle — rank resolving in every local contest while refusing to assemble globally. The consequence is the agenda. The amendment procedure votes pairs in a fixed order, the winner meeting the next entrant, and inside a cycle the procedure is decisive where the votes are not: the readout enumerates every possible order and counts the distinct winners a chair could seat. With a Condorcet winner it always reads 1; with the classic three-cycle it reads 3 — every member of the circle can be crowned.
What to try
Load the ladder (diversity 25%). Cycle frequency hugs 0% and agenda power reads 1 — all 24 agendas elect the same candidate, so procedure is harmless. Slide diversity to 100% and alternatives to 5: both readouts wake up.
Load the classic cycle. In the agenda machine, click candidates to send them last. Whoever enters the tournament last tends to win it: agenda power reads 3, and you can seat A, B or C without changing a single vote.
In engine 2, present A·B, B·C, then C·A. Each answer is decisive; together they circle. Search the mechanism for the number being maximised — there is none. Value from topology, in six units.
Committees, brackets, chairs
Organisations run on pairwise contests: motions against amendments, candidates against candidates, budget lines against budget lines. Where a genuine ladder exists, the order of business is bookkeeping — every route reaches the same summit. Where values circle, the order of business is the decision. The chair who "merely" schedules the motions, the clerk who drafts the order paper, the coach who seeds the bracket and the committee that designs the succession procedure hold power that never appears on the org chart — the org chart being the missing scalar itself, an official ladder the live pattern of deference declines to follow. It is also why genuinely plural societies cannot end their arguments by agreeing to "just rank the priorities": as the diversity dial shows, the more honestly plural the values in the room, the more often there is no ranking to find — only a procedure to fight over.
David Stark supplied the modern coda. Firms that keep several orders of worth alive at once — the engineers' ladder, the accountants' ladder, the designers' ladder, none permanently on top — are noisier day to day, and they can reorient when the environment flips which ladder matters. A pure hierarchy is coherent and captive: it can only ever do what its apex values, and when the world stops paying for that, the whole ladder falls together. Heterarchy, on this reading, is preparedness — dissonance kept deliberately in stock.
Next door
Heterarchy is one half of McCulloch's bequest to this cluster. The other is redundancy of potential command: authority should sit wherever the information is, migrating node to node — the command-side twin of value that refuses to centralise. Centralities makes the same point about importance: ask who matters most in a network and each measure returns a different ladder over the same web. And the minority game removes the fixed best at the level of action — what is worth doing depends on who else is doing it, and never settles.
The mapping
| In the model | In the world |
|---|---|
| an arrow i → j | One honest head-to-head: on this pair, the group truly has a view. |
| the hot cycle | Honest values that won't stack — every comparison sound, no ladder anywhere. |
| the agenda order | The meeting order, the tournament bracket, the succession procedure. |
| agenda power k | Why chairs, clerks and schedulers rule: inside a cycle, procedure chooses among k winners. |
| the missing u(·) | The org chart nobody's behaviour actually follows — official rank without a real scale. |
| the diversity dial | Why plural societies cannot simply "rank the priorities" — plurality breeds circles. |
| several live orders | Stark's coda: keeping more than one ladder alive on purpose buys the power to reorient. |
Where it tears
Cycle frequency depends on preference structure, and much of the time it is low. Black's theorem gives the clean case: when preferences are single-peaked along one shared dimension — everyone judging options by distance from their own point on a common line — majority preference is transitive and a Condorcet winner exists. The ladder preset shows the same effect through correlation. Heterarchy is a standing possibility that grows with plurality; the high-diversity numbers here are the extreme "impartial culture" case, an upper bound more than a portrait.
Agenda power as computed here assumes the chair knows every ranking and that members vote sincerely at each step. Under uncertainty the power shrinks — rig the order for a cycle that turns out not to exist and the same winner emerges by every route. And sophisticated voters who look ahead can vote strategically against the agenda, clawing part of the prerogative back. Real chairs rule at the margin, in the fog.
Stark's praise of heterarchy — several live orders of worth, faster reorientation — can excuse plain incoherence. Sometimes the cycle means the group has yet to do the hard work of deciding what it values, and celebrating "adaptability" merely defers the reckoning while the agenda-setters govern unnoticed. Both readings are true of different rooms; the instrument cannot tell you which room you are in.